Chromatic Matroids
On zeros of the characteristic polynomial of representable matroids of bounded tree-width.
Carolyn Chun § Rhiannon Hall § Criel Merino‡ Steven D. Noble§
§Departament of Mathematical Sciences Brunel University
‡Instituto de Matem´aticasUNAM, sede Oaxaca
Modern Techniques in Discrete Optimization: Mathematics, Algorithms and Applications, BIRS Oaxaca, 1-6 November 2015
Criel Merino On zeros of the characteristic polynomial of representable matroids of bounded tree-width. Chromatic Matroids Chromatic polynomial
The chromatic polynomial (for planar graphs) was first defined by Birkhoff in 1912 in an attempt to find an algebraic proof of the then open four-colour problem. The chromatic polynomial χ(G; λ) of a graph G counts, for every positive integer λ, the number of proper colouring of the vertices of G with λ colours. One can easily check that, if e is an edge of G, then
P(G, q) = P(G \ e, q) − P(G/e, q).
Criel Merino On zeros of the characteristic polynomial of representable matroids of bounded tree-width. Chromatic Matroids Chromatic polynomial
The following theorem is due to G. D. Birkhoff (and independently by H. Whitney1932) Theorem If G = (V , E) is a graph, then X χ(G; λ) = λκ(E) (−1)|A|λr(E)−r(A). (1) A⊆E
Criel Merino On zeros of the characteristic polynomial of representable matroids of bounded tree-width. Chromatic Matroids Chromatic polynomial
Since the chromatic polynomial can be evaluated at real and complex values, Birkhoff hoped to use analytic methods to prove the 4-colour conjecture. In fact, the four-colour Theorem is equivalent to stating that χ(G; 4) > 0 for all planar graphs G. Such an analytic proof was never found, but Birkhoff and Lewis in 1946 did show that if G is planar then χ(G; λ) > 0 for all real λ ≥ 5. They conjectured that 5 could be changed to 4, and almost 70 years later their conjecture is still open. Conjecture (Birkhoff-Lewis, 1946) If G is a planar graph, then χ(G; λ) > 0 for all real λ ≥ 4.
Criel Merino On zeros of the characteristic polynomial of representable matroids of bounded tree-width. Chromatic Matroids Chromatic polynomial
What about largest real roots? A family of graphs G has an upper root-free interval if there is a real number a such that no graph in G has a real root larger than a. The class of all graphs does not have an upper root-free interval (since you can get arbitrarily large chromatic roots from cliques).
Criel Merino On zeros of the characteristic polynomial of representable matroids of bounded tree-width. Chromatic Matroids Chromatic polynomial
This is different for a proper minor-closed class of graphs. Theorem (Mader ’67) For every k ∈ N there exists an integer f (k) such that any graph with minimum degree at least f (k) has a Kk -minor.
It follows that, for every proper minor-closed family of graphs G, there exists a smallest integer d = d(G) such that every graph in G has a vertex of degree at most d. Woodall and Thomassen proved the next result independently. Theorem (Thomassen ’97, Woodall ’97) If G is a proper minor-closed family of graphs, then (d(G), ∞) is an upper root-free interval for G.
Criel Merino On zeros of the characteristic polynomial of representable matroids of bounded tree-width. The idea of tree-width in graphs:
To say how “tree-like” the structure of a graph is.
tree decomposition
The width of a tree decomposition is the highest number of graph vertices contained in a node of the tree, minus one. The idea was extended to matroids by Hlineny and Whittle The idea was extended to matroids by Hlineny and Whittle
tree decomposition
Tree decomposition: • The nodes of � are “buckets”. • Each element of � goes in one bucket (partitioning �(�)). Buckets can be empty or contain many elements. Tree-Width for Matroids
• Given a tree decomposition, each node gets a weight called its node-width.
• The width of a tree decomposition is the highest value of its node-widths.
• The tree-width of a matroid is the minimum width of all its tree decompositions. Tree-Width for Matroids
• Given a tree decomposition, each node gets a weight called its node-width.
• The width of a tree decomposition is the highest value of its node-widths.
• The tree-width of a matroid is the minimum width of all its tree decompositions.
• How do we measure node-width? The width of a node � of � is
� � = � � − ∑ � � − � � � − �
= ∑� � � − � − � − 1 � � . The width of a node � of � is
� � = � � − ∑ � � − � � � − �
= ∑� � � − � − � − 1 � � .
What does this value mean geometrically? Brief Digression: A note on �-separations in representable matroids.
Let � be a matroid representable over ��(�). Then we can embed � in the projective space ��(� � − 1,�). Let (�, �) be a �-separation of �.
� �� � ∩ �� � � “guts” of (�, �)
plonk in points to get projective space here
Then we can add points of ��(� � − 1,�) to � into the guts of (�, �) to obtain matroid �′ and �-separation (� , � ) such that the guts of (� , � ) is the projective space ��(� − 2, �).
The width of a node � of � is
� � = � � − ∑ � � − � � � − �
= ∑� � � − � − � − 1 � � .
What does this value mean geometrically? If � is representable, then the width of node � is
� � = � � ∪ � ∪ � ∪ ⋯ ∪ � . If � is representable, then the width of node � is
� � = � � ∪ � ∪ � ∪ ⋯ ∪ � .
Note: If � is a leaf node of �, and � is the set of matroid elements in the �- bucket, then � � = � � . Example: Tree decompositions for � , and � ,
� 1 � , � 2
� � � � � � � � 1 2 3 3 3 3 2 1 node widths
� 1 � , � 2
� � � � � � � � 1 2 3 6 4 3 2 1 node widths Example: Tree decompositions for a daisy
� 3 3 �
ø � 3 5 � 3 node widths � 3 Example: Tree decompositions for a daisy
� 3 3 �
ø � 3 5
� 3 � 3
� 3 � 3
ø 3
3 � node widths ø 3 ø 3 � 3 � 3 The Chromatic Polynomial for Matroids
For a matroid M,
� �, � = (−1)| |� ( ). ⊆
The familiar deletion-contraction rules apply: • If � is not a loop or coloop then � �, � = � �\�, � − � �/�, � , • If � is a loop then � �, � = 0, • If � is a coloop then � �, � = � − 1 � �\�, � . Chromatic Matroids The Characteristic polynomial
For example,the characteristic polynomial of Um,n, 0 < m ≤ n, is Pm−1 k n m−k χUm,n (λ) = k=0 (−1) k (λ − 1).
For PG(r − 1, q), whose lattice of flats is isomorphic to the lattice of subspaces of the r-dimensional vector space over GF (q), has characteristic polynomial 2 r−1 χPG(r−1,q)(λ) = (λ − 1)(λ − q)(λ − q ) ··· (λ − q ).
Criel Merino On zeros of the characteristic polynomial of representable matroids of bounded tree-width. Chromatic Matroids generalized parallel connection
Given two matroids M1 = (E1, r1) y M2 = (E2, r2) we define the generalized parallel connection, PN (M1, M2), as the matroid over E1 ∪ E2 whose flats are the sets X de E1 ∪ E2 such that X ∩ E1 is a flat in M1 and X ∩ E2 is a flat in M2 and where ∼ ∼ N = M1|T = M2|T and T = E1 ∩ E2. Then,
χM1 (λ)χM2 (λ) χPN (M1,M2)(λ) = . (3) χN (λ)
Criel Merino On zeros of the characteristic polynomial of representable matroids of bounded tree-width. Theorem (Chun, H, Merino, Noble): Let � be a matroid representable over ��(�). If � has tree-width at most �, then there exists � ∈ ℝ such that � �, � > 0 for all � > � . Proof Outline:
For induction, lexicographically order all �� � -representable matroids by (rank, |� � |, something).
We make sure � is large enough that �� 0, � ,�� 1, � ,… , �� � − 1,� satisfy the theorem. Induction: Let � be �� � -representable with tree-width ≤ �. Assume all �� � -representable matroids of tree-width ≤ � that occur before � in the LEX ordering satisfy the theorem. Let � be an optimal tree decomposition of � and let (�, �) be a �′-separation of � displayed by � (note �′ ≤ �): Let {� , � , … , � } be the elements of �� �(�) − 1, � in the guts of �, � .